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December 9, 2010
number theoryrelatively primenumber theory unsolved

Problem Statement

For each n∈Nn\in \mathbb{N}, let S(n)S(n) be the sum of all numbers in the set {1,2,3,⋯ ,n}\{ 1, 2, 3, \cdots , n \} which are relatively prime to nn.
(a)(a) Show that 2⋅S(n)2 \cdot S(n) is not a perfect square for any nn.
(b)(b) Given positive integers m,nm, n, with odd nn, show that the equation 2⋅S(x)=yn2 \cdot S(x) = y^n has at least one solution (x,y)(x, y) among positive integers such that m∣xm|x.
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